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discussed in biography

...Out (1966) was a charmless Jerry Lewis vehicle. Douglas later made three hard-boiled Sinatra films:
Tony Rome (1967) and its sequel
Lady in Cement (1968) and (arguably the best of the trio)
The Detective (1968), which featured the notable cast of Robert Duvall, Lee Remick, Ralph Meeker, and Jack...

“Tony Rome”

...hotel, where Sinatra performed as a singer at night after filming during the day. Sinatra’s daughter Nancy sang the film’s eponymous title song. An entertaining sequel,
Lady in Cement (1968), again featured Sinatra in the lead role and costarred Raquel Welch.